A note on Pontryagin duality and continuous logic

Abstract

We exhibit Pontryagin duality as a special case of Stone duality in a continuous logic setting. More specifically, given an abelian topological group A, and F the family (group) of continuous homomorphisms from A to the circle group T, then, viewing (A,+) equipped with the collection F as a continuous logic structure M, we show that the local type space SF(M) is precisely the Pontryagin dual of the group F where the latter is considered as a discrete group.We conclude, using Pontryagin duality (between compact and discrete abelian groups), that SF(M) is the Bohr compactification of the topological group A.